Title Transmission of ultrasonic waves at oblique incidence tocomposite laminates with spring-type interlayer interfaces.

Author(s) Ishii, Yosuke; Biwa, Shiro

Citation The Journal of the Acoustical Society of America (2015),138(5): 2800

Issue Date 2015-11-05

URL http://hdl.handle.net/2433/207703

Right

© 2015 Acoustical Society of America. This article may bedownloaded for personal use only. Any other use requires priorpermission of the author and the Acoustical Society ofAmerica. The following article appeared in [J. Acoust. Soc.Am. 138, 2800 (2015)] and may be found athttp://scitation.aip.org/content/asa/journal/jasa/138/5/10.1121/1.4934265.

Type Journal Article

Textversion publisher

Kyoto University

Transmission of ultrasonic waves at oblique incidence tocomposite laminates with spring-type interlayer interfaces

Yosuke Ishii and Shiro Biwaa)

Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Katsura,Nishikyo-ku, Kyoto 615-8540, Japan

(Received 2 April 2015; revised 5 August 2015; accepted 28 September 2015; published online 5November 2015)

The transmission characteristics of ultrasonic waves at oblique incidence to composite laminates

are analyzed theoretically by the stiffness matrix method. The analysis takes into account the pres-

ence of thin resin-rich regions between adjacent plies as spring-type interfaces with normal and

shear stiffnesses. The amplitude transmission coefficient of longitudinal wave through a unidirec-

tional laminate immersed in water is shown to be significantly influenced by the frequency, the

interlayer interfacial stiffnesses, and the incident angle. Using Floquet’s theorem, the dispersion

relation of the infinitely extended laminate structure is calculated and compared to the transmission

coefficient of laminates of finite thickness. This reveals that the ranges of frequency and interfacial

stiffnesses where the Floquet waves lie in the band-gaps agree well with those where the transmis-

sion coefficient of the finite layered structure is relatively small, indicating that the band-gaps

appear even in the laminate with a finite number of plies. The amplitude transmission coefficient

for an 11-ply carbon-epoxy unidirectional composite laminate is experimentally obtained for vari-

ous frequencies and incident angles. The low-transmission zones observed in the experimental

results, which are due to the critical angle of the quasi-longitudinal wave and the Bragg reflection,

are shown to be favorably compared with the theory. VC 2015 Acoustical Society of America.

[http://dx.doi.org/10.1121/1.4934265]

[TK] Pages: 2800–2810

I. INTRODUCTION

Fiber-reinforced composite laminates have been exten-

sively used in aerospace, automotive, marine, and civil engi-

neering since they have the advantages of high specific

stiffness, high specific strength, low coefficient of thermal

expansion, etc., compared to monolithic materials.

Understanding the wave propagation behavior in such struc-

tures is important from the viewpoints of the nondestructive

characterization of their property as well as their damage di-

agnosis to ensure the structural safety and reliability.

The ultrasonic wave propagation in composite laminates

has been widely studied and various methods have been pro-

posed for the nondestructive evaluation of material proper-

ties.1 The anisotropic elastic constants of a composite plate

can be evaluated by immersing it in fluid and measuring the

velocity for three types of transmitted bulk waves at various

angles of incidence, i.e., the quasi-longitudinal, fast, and

slow quasi-transverse waves.2–6 In order to remove the influ-

ence of beam refraction at the top and bottom surfaces of

specimen, the so-called double through transmission tech-

nique was proposed.7,8 Extending this procedure, Chu and

Rokhlin9 evaluated the elastic constants of unidirectionally

reinforced plies constituting a cross-ply laminate. Similar

measurements for other stacking sequences were performed

by Hosten,10 and those using the air-coupled transducers

were also reported by Hosten et al.11 In addition to the ve-

locity measurement, the imaginary part of complex elastic

constants of a unidirectional composite plate can be eval-

uated as well from the reflected or transmitted amplitude of

bulk waves,12–14 and the temperature dependence of such

properties was investigated by Baudouin and Hosten.15

These velocity- or amplitude-based evaluation methods are

applicable for sufficiently thick plates where the transmitted

bulk modes can be separated in the time domain. Evaluation

procedures for the elastic constants of composite laminates

for a wider range of their thickness have also been devel-

oped, using the amplitude transmission spectrum16 or the

dispersion curve in the frequency-wavenumber domain.17

The above-mentioned studies assumed that all plies of

the laminate were perfectly bonded and the effects of inter-

layer interfaces were ignored. Most resin-based composite

laminates such as carbon-epoxy and glass-epoxy laminates,

however, have resin-rich regions with a few microns thick-

ness between adjacent plies and such imperfections have sig-

nificant influence on the mechanical performance of

laminated structures;18,19 hence, the nondestructive evalua-

tion of the soundness of interlayer interfaces is important. To

this end, it is essential to understand the influence of such

regions upon the ultrasonic wave propagation behavior.

One way to consider the effect of resin-rich regions on

the wave propagation in composite laminates is to model

them as a thin resin layer of finite thickness between neigh-

boring plies. For example, Wang and Rokhlin20 demon-

strated that the resin-rich regions have a significant effect

on the reflection characteristics and band-gap behavior of

ultrasonic waves at normal incidence to multidirectional

composite laminates. Another possible approach is to usea)Electronic mail: biwa@kuaero.kyoto-u.ac.jp

2800 J. Acoust. Soc. Am. 138 (5), November 2015 VC 2015 Acoustical Society of America0001-4966/2015/138(5)/2800/11/$30.00

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spring-type interfaces:21–27 When performing the wave prop-

agation analysis for adhesive bonds or other imperfect inter-

faces, the interfacial region much thinner than the

wavelength of the ultrasound can be modeled as a spring-

type interface with finite interfacial normal and shear stiff-

nesses. On the basis of this idea, the influence of random

deviations in the interlayer interfacial normal stiffness or the

ply wave velocity on the propagation behavior of the longi-

tudinal wave impinging perpendicularly on a composite lam-

inate was analyzed by Lu and Achenbach28 and Lu.29 Ishii

and Biwa30,31 studied the influence of interfacial normal

stiffness on the wave propagation and band-gap behavior

theoretically, and showed that the interfacial normal stiffness

can be evaluated from the extremal frequencies of amplitude

reflection spectrum. The above-mentioned works20,28–31 are,

however, limited to the simplest case of the normal inci-

dence of longitudinal wave. The effect of resin-rich regions

on the wave propagation behavior of composite laminates

for arbitrary angles of incidence remains as a subject to be

explored in further depth.

In the present study, the ultrasonic wave propagation in

composite laminates with spring-type interlayer interfaces at

oblique incidence is analyzed theoretically by using the

stiffness-matrix approach.32,33 In particular, the influence of

the frequency, the interlayer interfacial stiffnesses, and the

angle of incidence on the wave transmission and band-gap

behavior is elucidated. The present analysis is, however,

focused on unidirectional lay-ups for which the interlayer

interfaces are solely responsible for the band-gap formation.

This paper is constructed as follows: In Sec. II, the formula-

tion of the stiffness-matrix method is briefly summarized

and the amplitude transmission coefficient of a unidirec-

tional composite laminate immersed in water is calculated

for various frequencies, interfacial normal and shear stiff-

nesses, and angles of incidence. Using Floquet’s theo-

rem34–36 combined with the stiffness matrix, the dispersion

relation of the infinitely extended laminate structure is calcu-

lated and compared to the transmission characteristics of

laminates of finite thickness in Sec. III. Such a comparison

between finite and infinite laminated structures has been con-

sidered by Wang and Rokhlin20 for multidirectional compos-

ite laminates. In the present study, the generation behavior

of band-gaps of Floquet waves is examined for arbitrary

propagation directions in the presence of spring-type inter-

layer interfaces in the unidirectional laminate. Comparison

between experimental and theoretical transmission coeffi-

cients for an 11-ply carbon-epoxy composite laminate is

shown in Sec. IV, and the conclusion of this study is sum-

marized in Sec. V.

II. ANALYSIS OF WAVE PROPAGATION BEHAVIOR INA UNIDIRECTIONAL COMPOSITE LAMINATE WITHSPRING-TYPE INTERLAYER INTERFACES ATOBLIQUE INCIDENCE

A. Computation of reflection/transmission coefficientby stiffness matrix method

Consider a unidirectional composite laminate consisting

of N anisotropic elastic plies and (N–1) spring-type

interlayer interfaces as shown in Fig. 1. If the harmonic

plane longitudinal wave with unit amplitude impinges

obliquely on the laminate immersed in ideal fluid from

x3>Z0 with the direction determined by the angle of inci-

dence h as well as the plane of propagation u, then the dis-

placements of incident, reflected, and transmitted waves can

be written as

uinc ¼sin h sin /

�sin h cos /

�cos h

0B@

1CA

� ei½kffx1 sin h sin /�x2 sin h cos /�ðx3�Z0Þ cos hg�xt�

ðx3 > Z0Þ; (1)

uref ¼ R

sin h sin /

�sin h cos /

cos h

0B@

1CA

� ei½kffx1 sin h sin /�x2 sin h cos /þðx3�Z0Þ cos hg�xt�

ðx3 > Z0Þ; (2)

utra ¼ T

sin h sin /

�sin h cos /

�cos h

0B@

1CA

� ei½kffx1 sin h sin /�x2 sin h cos /�ðx3�Z0Þ cos hg�xt�

ðx3 < ZNÞ; (3)

respectively, where x (¼2pf) is the angular frequency and kf

is the wavenumber in the fluid. The complex reflection and

transmission coefficients R and T can be calculated by the

stiffness matrix method32,33 as

FIG. 1. (Color online) A unidirectional composite laminate with spring-type

interlayer interfaces immersed in ideal fluid.

J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa 2801

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R ¼ cþ SG33

� �cþ SG

66

� �� SG

36SG63

c� SG33

� �cþ SG

66

� �þ SG

36SG63

;

T ¼ �c2SG63

c� SG33

� �cþ SG

66

� �þ SG

36SG63

e�ikfH cos h;

c ¼ cos hiqfVfx

; (4)

where qf and Vf¼x/kf are the density and wave speed of the

surrounding fluid and H¼ Z0 � ZN is the laminate thickness.

In the above expression, SijG are the components of the

inverse of 6� 6 global stiffness matrix which is obtained by

applying the recursive algorithm32 to all of the local stiffness

matrices of the ply Kplym ðm ¼ 1; 2;…;NÞ as well as the inter-

layer interface Kspm ðm ¼ 1; 2;…;N � 1Þ defined as

rjx3¼Zm�1;�

rjx3¼Zm;þ

!¼ Kply

m

�ujx3¼Zm�1;�

ujx3¼Zm;þ

�;

rjx3¼Zm;þ

rjx3¼Zm;�

!¼ Ksp

m

ujx3¼Zm;þ

ujx3¼Zm;�

!;

r ¼r13

r23

r33

0B@

1CA; u ¼

u1

u2

u3

0B@

1CA; (5)

where x3¼Zm (m¼ 1, 2,…, N � 1) correspond to the loca-

tion of interlayer interfaces, and the subscript “þ” (“�”)

denotes that the corresponding coordinate is approached

from the positive (negative) x3 side. In the present analysis,

the matrices Kplym and Ksp

m are assumed to be the same for all

plies and interfaces. The latter can be calculated by using the

interfacial normal and shear stiffnesses, KN, KT1, and KT2, as

Kspm ¼

KT1 0 0 �KT1 0 0

0 KT2 0 0 �KT2 0

0 0 KN 0 0 �KN

KT1 0 0 �KT1 0 0

0 KT2 0 0 �KT2 0

0 0 KN 0 0 �KN

266666664

377777775: (6)

The above expression is valid to model the interlayer imper-

fections of unidirectional composite laminates where the fiber

direction of all plies is identical with x1 (or x2) direction.

Other stacking sequences such as angle-ply, cross-ply, or

quasi-isotropic lay-up can be modeled by including additional

non-zero terms which express the coupling effects24 in Eq. (6)

and allow the matrix components to vary for different interfa-

ces. For the details of the calculation of Kplym and the recursive

algorithm mentioned above, refer to Refs. 32 and 33. It should

be noted that the stiffness matrix approach can compute the

reflection and transmission coefficients more stably than the

transfer matrix approach37–40 which encounters the numerical

instability issues for high frequency or large ply thickness.

B. Influence of interlayer interfacial stiffness and angleof incidence on amplitude transmission spectrum

Using the approach mentioned above, the amplitude

transmission spectra of an 11-ply unidirectional composite

laminate immersed in water are calculated for various inter-

layer interfacial stiffnesses and angles of incidence with the

material properties in Table I, and depicted in Figs. 2 and 3.

The unidirectionally reinforced plies are modeled as trans-

versely isotropic elastic media and stacked so that the fiber

direction is identical with the x1 direction for all plies. Note

that the imaginary parts of ply stiffness shown in Table I are

neglected throughout the numerical analysis in Secs. II and

III, but are taken into account in the comparison between

experiment and theory in Sec. IV. The angle u which repre-

sents the deviation of the plane of propagation from the iso-

tropic plane of ply (x2 � x3 plane) is fixed as 45�, and for

simplicity of analysis, it is assumed that the interfacial shear

stiffness is orientation-independent, i.e., KT1¼KT2 � KT.

The horizontal axis in Figs. 2 and 3, whose range corresponds

to approximately 0� f� 16 MHz, is normalized by the ply

thickness and the longitudinal wave velocity of the ply in the

x3 direction, while the vertical ones, whose ranges correspond

to 0.08�KN� 8 MPa/nm and 0.04�KT� 4 MPa/nm, respec-

tively, are normalized by the ply thickness and stiffness.

It is seen in Figs. 2 and 3 that the transmission coeffi-

cient is influenced by not only the frequency and interfacial

stiffnesses but also the angle of incidence. This is because

the number of propagating bulk modes existing in the lami-

nate changes according to the latter parameter. Wang and

Rokhlin20 examined this variation by invoking the critical

angles of Floquet waves for multidirectional laminates.

Since the present analysis focuses on unidirectional lami-

nates, the critical angles of Floquet waves can be reasonably

approximated by those of bulk waves in the ply. In Fig. 4,

the critical angles of the quasi-longitudinal and two quasi-

transverse waves at an interface between water and the ply

are shown as a function of the plane of propagation u.

Because of the anisotropic nature of the ply, the critical

angles are dependent on u and they take 16�, 36�, and 56� at

u¼ 45�. In the case of normal incidence [Figs. 2(a) and

3(a)], only the pure longitudinal mode propagates in the lam-

inate in the direction normal to the interfaces, so the trans-

mission coefficient is influenced only by the normal

component of the interlayer interfacial stiffness. It is also

seen that the transmission coefficient oscillates with the fre-

quency due to the finite number of plies.30 As the angle of

TABLE I. Material properties of the composite laminate and the surround-

ing fluid (water).

Stacking sequence of laminate [0]11

Complex elastic constants

of ply (transverse isotropy

with x1 as fiber direction is assumed.)

C11 81.5 – 2.90i GPa

C13 4.9 – 0.33i GPa

C33 15.2 – 0.29i GPa

C44 3.2 – 0.14i GPa

C66 6.9 – 0.24i GPa

Density of ply q 1.5� 103 kg/m3

Thickness of ply h 0.19 mm

Density of fluid qf 1.0� 103 kg/m3

Wave speed in fluid Vf 1.5� 103 m/s

2802 J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa

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incidence increases from zero [Figs. 2(b) and 3(b)], two other

bulk modes come to propagate in the plies. In this case, the

quasi-longitudinal as well as two quasi-transverse modes in

general have the displacement components parallel to the

interface. Hence, the interfacial shear stiffness is also involved

and the transmission coefficient exhibits the more complicated

behavior than the normal incidence case. In Figs. 2(c) and

3(c), where the angle of incidence is above the critical angle

of quasi-longitudinal wave, the number of bulk modes in the

plies reduces to two, but the transmission characteristics

remain complex. When the angle of incidence further

increases so as to exceed the next critical angle of fast quasi-

transverse wave, only the slow quasi-transverse mode can

propagate in the laminate in the thickness direction, and the

patterns of the transmission coefficient shown in Figs. 2(d)

and 3(d) become simpler as in the case of normal incidence

where the transmission characteristics are governed by a sin-

gle bulk mode.

Furthermore, it is recognized that there are certain fre-

quency ranges in Figs. 2 and 3 where the transmission coeffi-

cient falls to a low level. For example, two frequency bands of

vanishingly small transmission coefficient are seen in Fig. 2(a)

spanning approximately for 0.28� fh(q/Re[C33])1/2� 0.50 and

0.65� fh(q/Re[C33])1/2� 1 when KNh/Re[C33]¼ 1. Their loca-

tion and bandwidth are, as a general trend, influenced by the

interfacial stiffnesses and the angle of incidence.

III. DISPERSION RELATION OF INFINITELY PERIODICSTRUCTURE

In this section, the generation of low-transmission fre-

quency ranges found Sec. II B is investigated in more detail

based on the dispersion relation of the infinitely periodic

structure.

A. Computation of Floquet wavenumber

Consider a periodic structure shown in Fig. 5, which has

an infinite number of unit-cells each consisting of a unidirec-

tional ply and a spring-type interface. The stiffness matrix

for a unit-cell K is given as

rjx3¼Zm�1;þ

rjx3¼Zm;þ

!¼ K

ujx3¼Zm�1;þ

ujx3¼Zm;þ

!; K �

K11 K12

K21 K22

� �;

(7)

where Kij (i, j¼ 1, 2) represent 3� 3 matrices. For such a

periodic system, Floquet’s theorem yields the so-called peri-

odic relation

rjx3¼Zm�1;þ

ujx3¼Zm�1;þ

!¼ exp ðifhuÞ

rjx3¼Zm;þ

ujx3¼Zm;þ

!; (8)

FIG. 2. (Color online) The variation of amplitude transmission coefficient with the normalized frequency and the normalized interlayer interfacial normal stiff-

ness for four different angles of incidence with fixed interfacial shear stiffness KT¼ 0.04 MPa/nm (KTh/Re[C66]¼ 1) and plane of propagation u¼ 45�.

FIG. 3. (Color online) The variation of amplitude transmission coefficient with the normalized frequency and the normalized interlayer interfacial shear stiff-

ness for four different angles of incidence with fixed interfacial normal stiffness KN¼ 0.8 MPa/nm (KNh/Re[C33]¼ 10) and plane of propagation u¼ 45�.

J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa 2803

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where f is the x3 component of the Floquet wavenumber34–36

and hu¼ Zm�1,þ � Zm,þ (¼ h) is the thickness of unit-cell

(the theorem is also referred to as the Bloch theorem but the

nomenclature commonly used in many studies for composite

laminates34–36 is followed here). From Eqs. (7) and (8), the

characteristic equation for the Floquet wave in the infinitely

periodic structure can be written as

A3 cos ð3fhuÞ þ A2 cos ð2fhuÞ þ A1 cos ðfhuÞ þ A0 ¼ 0;

(9)

where

A3 ¼ detðK21Þ; (10)

A2 ¼1

2det K22 �K11 þK21ð Þ�þ det K22 �K11 �K21ð Þg � det K22 �K11ð Þ;

(11)

A1 ¼1

2det K22 �K11 þK21ð Þ þ det K22 �K111K12ð Þ�þ det K21 �K12ð Þg � 2det K21ð Þ;

(12)

A0 ¼1

4det K22 �K11 þK12 �K21ð Þ�þ det K22 �K11 �K12þK21ð Þg � A2: (13)

Equation (9) has six solutions for f in the range of �p/hu

�Re[f]�p/hu, which satisfy the relation f1¼�f4, f2¼�f5,

and f3¼�f6, i.e., they correspond to three pairs of Floquet

waves propagating in the opposite x3 directions. This charac-

teristic equation was derived by Wang and Rokhlin,36 but

the correct expressions of its coefficients have been reder-

ived here as shown in Eqs. (10)–(13). From Eq. (9), f can be

given as

f ¼ 2mpþ arg Xð Þhu

� ilnjXj

hu

; (14)

where m is an integer, arg(•) and j•j represent the argument

and modulus of a complex number, and X (¼ exp(ifhu)) is a

root of

X2 � gX þ 1 ¼ 0; (15)

where g is a root of the following cubic equation:

A3g3 þ A2g

2 þ ðA1 � 3A3Þgþ 2ðA0 � A2Þ ¼ 0: (16)

In what follows, the Floquet waves propagating in the

positive x3 direction are only considered, whose wavenum-

bers are denoted by f1, f2, and f3. The relation between the

x3 component of the normalized Floquet wavenumbers and

the normalized frequency for the laminate parameters given

in Table I (imaginary parts of ply stiffness are neglected as

mentioned in Sec. II B) is illustrated in Fig. 6 for different

propagation directions expressed in terms of the correspond-

ing incident angle from water. The real part of the wavenum-

bers is drawn with the so-called extended zone scheme,34

where the integer m in Eq. (14) is determined so that the dis-

persion curve becomes continuous. The angle for plane of

propagation u is fixed again as 45� and the normalized inter-

layer interfacial stiffnesses as KNh/Re[C33]¼ 12.5 and KTh/

Re[C66]¼ 2.8, which correspond to KN¼ 1 MPa/nm and

KT¼ 0.1 MPa/nm, respectively.

The Floquet wave is a linear combination of three types

of classical plane waves35 which propagate in the ply, i.e.,

one quasi-longitudinal and two quasi-transverse waves. With

the stacking sequence concerned here, however, each

Floquet wave is dominated by one of them. In the case of

normal incidence or in the limiting case of KN¼KT¼1(perfect bonding), each Floquet wave consists of a single

classical plane wave. For this reason, three Floquet waves in

Fig. 6 are denoted by referring to the corresponding domi-

nant plane wave modes.

It is seen in Fig. 6 that when the frequency increases

from zero, the real parts of the Floquet wavenumber also

FIG. 4. (Color online) The critical angle of three bulk modes at the water-

ply interface.

FIG. 5. An infinitely laminated composite and its unit-cell.

2804 J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa

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increase from zero, and in certain frequency ranges they

possess non-zero imaginary parts. These ranges are the

so-called frequency band-gaps (or stopping bands), which

indicate that the corresponding mode cannot propagate in

the periodic structure since the Bragg reflection occurs. Such

band-gaps appear when the following condition is satisfied:

Re½fa�hu=pþ Re½fb�hu=p ¼ 2n; ða; b ¼ 1; 2; 3Þ; (17)

where n stands for an integer. The derivation of the above

relation is given in the Appendix. When a¼b in Eq. (17),

the Bragg reflection occurs with a single Floquet wave,

which results in the single band-gaps indicated in Fig. 6 by

“L,” “T1,” “T2,” “QL,” “QT1,” and “QT2.” On the other

hand, when a 6¼ b, two Floquet waves suffer the Bragg

reflection in combination and the resulting double band-gaps

are indicated by “QL-QT1,” “QL-QT2,” and “QT1-QT2.”

When h¼ 0� [Fig. 6(a)], three Floquet waves are iden-

tical with the pure longitudinal (L) and transverse (T1 and

T2) modes as mentioned above. Since such modes do not

couple with one another, i.e., no mode conversion occurs at

the interlayer interfaces, there exist only single band-gaps

in Fig. 6(a). When h¼ 10� [Fig. 6(b)], the Floquet waves

consist of different classical plane waves which undergo

the mode conversion at every interface, so not only the sin-

gle but also double band-gaps are generated. This is also

true for h¼ 20� [Fig. 6(c)] except that the Floquet wave

dominated by the quasi-longitudinal mode, which propa-

gates in the direction parallel to the interfaces, has no influ-

ence on the generation of band-gaps. When h¼ 40� [Fig.

6(d)], the wavenumbers of two faster Floquet waves

become pure imaginary, and the band structure is deter-

mined only by the remaining wave dominated by the slow

quasi-transverse mode.

B. Comparison with transmission characteristicsof finite layered structure

Using the stiffness matrix method again, the amplitude

transmission spectra of the immersed composite laminate are

calculated for different number of plies with fixed interfacial

stiffnesses KN¼ 1 MPa/nm and KT¼ 0.1 MPa/nm, angle of

incidence h¼ 10�, and plane of propagation u¼ 45�, and

shown in Fig. 7. The low-transmission frequency ranges cen-

tered at around fh(q/Re[C33])1/2¼ 0.43 and 0.87 become

more distinct as the number of plies increases, and such

regions are in good agreement with the band-gaps of the cor-

responding infinitely layered structure in Fig. 6(b). This indi-

cates that the band-gaps appear even in the finite layered

laminate and becomes remarkable with the increasing num-

ber of plies. It is also found that the band-gaps of the Floquet

waves dominated by quasi-transverse modes seen in Fig.

6(b) do not form clear low-transmission ranges in Fig. 7.

FIG. 6. (Color online) The dispersion

relation of Floquet waves for four differ-

ent corresponding angles of incidence

from water with fixed interlayer interfa-

cial normal stiffness KN¼ 1 MPa/nm,

shear stiffness KT¼ 0.1 MPa/nm, and

plane of propagation u¼ 45�.

FIG. 7. The variation of the amplitude transmission spectrum with the num-

ber of plies for fixed interlayer interfacial normal stiffness KN¼ 1 MPa/nm,

shear stiffness KT¼ 0.1 MPa/nm, angle of incidence h¼ 10�, and plane of

propagation u¼ 45�.

J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa 2805

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This is because the wave field in the finite layered laminate

is mainly governed by the quasi-longitudinal mode; namely,

even if the Floquet waves dominated by the quasi-transverse

modes lie in the band-gaps, the ultrasonic wave can transmit

through the laminate as the quasi-longitudinal mode.

With the same parameters used for the computation of

Figs. 2 and 3, the number of propagating Floquet waves is

calculated and shown in Figs. 8 and 9 as functions of the nor-

malized frequency, the normalized interlayer interfacial stiff-

nesses, and the corresponding angle of incidence from water.

Note that for h¼ 0� in Figs. 8(a) and 9(a), only the pure lon-

gitudinal mode is counted in order to compare with Figs.

2(a) and 3(a) as it is the only wave that can couple with

water. It is seen that if the number of propagating Floquet

waves becomes zeros (white zones) in Figs. 8 and 9, then the

transmission coefficients drops to a low level in Figs. 2 and

3, which reconfirms the results mentioned in the last para-

graph. Its converse is not necessarily true, however, particu-

larly in the cases of h¼ 10� and 20�. In the ranges of

approximately 0.70� fh(q/Re[C33])1/2� 0.76 and 1�KNh/

Re[C33]� 3 for h¼ 10�, the Floquet wave dominated by the

slow quasi-transverse mode is propagative as indicated by

“QT2” in Fig. 8(b), but the transmission coefficient becomes

very low in Fig. 2(b). This is, as also mentioned above,

because the transmission coefficient is mainly governed by

one dominant classical plane wave mode (quasi-longitudinal

for h¼ 10� and fast quasi-transverse for h¼ 20�) and not

sensitive to whether the other modes can transmit through

the laminate or not. In addition to that, when h¼ 40�, there

exists a frequency range at around fh(q/Re[C33])1/2¼ 0.02

where the transmission coefficient takes a relatively small

value in Figs. 2(d) and 3(d) but no band-gaps can be seen in

Figs. 8(d) and 9(d). This is likely due to the finite thickness

of the laminate, i.e., an eigen-vibration of laminate is excited

and the reflection becomes remarkable.

Although the plane of propagation has been fixed as

u¼ 45� in this analysis, the transmission characteristics for

other values of u can be explained in the same manner as

described above.

IV. COMPARISON WITH EXPERIMENT

The double-through transmission measurement7,8 was

carried out for an 11-ply composite laminate with unidirec-

tional stacking sequence. The specimen, which was supplied

by Mitsubishi Rayon, Co. Ltd., Japan, was made of TR30

carbon fibers of about 7 lm diameter and #340 epoxy resin,

and had the ply thickness of about 0.19 mm. The ultrasonic

waves transmitted through the specimen twice were meas-

ured for different angles of incidence h as well as planes

FIG. 8. (Color online) The variation of the number of propagating Floquet waves with the normalized frequency and the normalized interfacial normal stiff-

ness for four different corresponding angles of incidence from water with fixed KT¼ 0.04 MPa/nm (KTh/Re[C66]¼ 1) and plane of propagation u¼ 45�.

FIG. 9. (Color online) The variation of the number of propagating Floquet waves with the normalized frequency and the normalized interfacial shear stiffness

for four different corresponding angles of incidence from water with fixed KN¼ 0.8 MPa/nm (KNh/Re[C33]¼ 10) and plane of propagation u¼ 45�.

2806 J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa

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of propagation u by using a setup shown in Fig. 10, which

consists of a personal computer (MATLAB), an Agilent digital

oscilloscope DSO5034A, a Panametrics pulser-receiver

5072PR, a motor controller, two five-phase stepping motors

with angle resolutions 0.004� for both h and u, a piezoelec-

tric broadband transducer with nominal frequency 15 MHz

(IT1506R, Insight, Inc., Japan), and an acoustic mirror made

of a polished stainless steel disk with 50 mm diameter and

30 mm thickness. The signals were measured with a gate

length of 10 ls, which was chosen in order to include not

only the first arrival waves but also the following ones due to

the multiple reflections inside the specimen but not to cap-

ture the waves reflected from the bottom of the mirror (it

took about 10 ls for the longitudinal wave to make a round

trip in the mirror). This gate length was confirmed to be suf-

ficient for the present specimen as the amplitude of arrival

waves to be measured dropped to the noise level in at most

6 ls after the first arrival wave. The received signals were

digitized at the sampling frequency of 100 MHz and aver-

aged 100 times. The measured waveforms were then zero

padded to contain 4096 points (resulting frequency incre-

ment was 0.02 MHz), Fourier transformed without window-

ing, and normalized by the spectrum of the reference wave

measured without the specimen so as to compute the trans-

mission coefficients.

The variation of the obtained transmission coefficient

with the frequency and the angle of incidence for three types

of planes of propagation (u¼ 0� corresponds to the plane

whose normal coincides with the fiber direction) is depicted

in Fig. 11 for a finite frequency range of 5 to 15 MHz due to

the limited bandwidth of the transducer. It is noted that

obtained by this measurement is the squared amplitude trans-

mission coefficient since the doubly transmitted wave is

detected. The theoretical results calculated by the stiffness

matrix method are also shown in Fig. 11 for comparison,

where the properties in Table I including the imaginary parts

of ply stiffness are used with the interlayer interfacial stiff-

nesses KN¼ 3.0 MPa/nm and KT¼ 0.8 MPa/nm. These pa-

rameters are chosen so that the theoretical transmission

coefficients fit the experimental ones in Figs. 11(a)–11(c) as

FIG. 10. (Color online) Experimental apparatus.

FIG. 11. (Color online) Experimental

and theoretical squared amplitude

transmission coefficients of an 11-ply

carbon-epoxy unidirectional composite

laminate for various frequencies and

angles of incidence with three different

planes of propagation.

J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa 2807

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closely as possible. The values for the components of com-

plex ply stiffness and the interfacial normal stiffnesses are

representative for the unidirectional carbon-fiber-reinforced

composite.31,40 Although there are no literatures available

for the interfacial shear stiffness of composite laminates, the

above-mentioned value appears reasonable from the compar-

ison of interfacial stiffness ratio KT/KN¼ 0.27 with that

measured for a thin epoxy adhesive layer between two glass

plates 0.23.25

The low-transmission zones seen over the wide fre-

quency ranges at around h¼ 30� in Fig. 11(a), h¼ 10� in

Fig. 11(b), and h¼ 8� in Fig. 11(c) are due to the critical

angle of the quasi-longitudinal wave, while those seen in the

limited frequency ranges such as at about 8 MHz in Fig.

11(c) are due to the Bragg reflection. It is found that these

features are well reproduced by the theory. It should be

noted that the latter feature cannot be described unless the

influence of interlayer interfaces is incorporated. Besides the

locations of these low-transmission zones, fine oscillatory

patterns of the amplitude transmission coefficient in the

plane of frequency and incident angle are also reproduced by

the theory with the interfacial stiffness parameters as identi-

fied above.

V. SUMMARY

In this paper, the transmission characteristics of ultra-

sonic waves in unidirectional composite laminates with

spring-type interlayer interfaces at oblique incidence have

been analyzed theoretically by using the stiffness matrix

method. It has been shown that the amplitude transmission

coefficient is influenced by the frequency, the interlayer

interfacial normal and shear stiffnesses, and the angle of

incidence. The dispersion relation of the corresponding infin-

itely periodic structure has been calculated using Floquet’s

periodic condition and compared to the transmission coeffi-

cient of the finite layered structure. As a result, the ranges of

the frequency and the interfacial stiffnesses where the

Floquet waves lie in the band-gaps are in good agreement

with the low-transmission zones of the finite layered case,

indicating that the ultrasonic wave is prevented from trans-

mitting through the laminate due to the Bragg reflection. The

amplitude transmission coefficient for an 11-ply carbon-ep-

oxy composite laminate with unidirectional stacking

sequence has been experimentally obtained for various fre-

quencies and angles of incidence. It has been shown that the

observed low-transmission zones caused by the critical angle

of quasi-longitudinal wave as well as the Bragg reflection

are favorably compared with the theory.

The results obtained in this analysis can be helpful when

evaluating the quality of the interlayer interfaces of compos-

ite laminates from the propagation characteristics of bulk

waves. The present study has focused on the laminate with

unidirectional lay-up where the reflection and transmission

at the interlayer interfaces are due only to the finite interfa-

cial stiffnesses. For other stacking sequences, e.g., angle-ply,

cross-ply, or quasi-isotropic laminates, the mismatch of the

acoustic impedances between neighboring plies also has an

influence on the scattering at the interfaces as discussed in

Ref. 20. In such situations, more complicated generation

behavior of band-gaps will be observed, which remain as

issues for the future study.

ACKNOWLEDGMENTS

This work has been supported by JSPS KAKENHI

Grant Nos. 25-1754 and 25289005.

APPENDIX: DERIVATION OF EQ. (17)

Assuming that the x3 component of the Floquet wave-

number is given as

f ¼ ð2mpþ aþ ibÞ=hu; (A1)

where m is an integer and �p< a�p and b are real num-

bers, Eq. (15) can be rewritten as

coshb cos a� isinhb sin a ¼ g2: (A2)

The band structure of the Floquet waves can be divided into

four cases according to g as follows:

ðaÞ jRe½g�j < 2 and Im½g� ¼ 0; (A3)

ðbÞ jRe½g�j ¼ 2 and Im½g� ¼ 0; (A4)

ðcÞ jRe½g�j > 2 and Im½g� ¼ 0; (A5)

ðdÞ Im½g� 6¼ 0: (A6)

In the case of (a), Eq. (A2) is reduced to

a ¼ 6cos�1Re g½ �

2

� �; (A7)

b ¼ 0: (A8)

In Eq. (A7), the plus-minus sign represents the pair of

Floquet waves propagating in the opposite x3 direction.

Since the Floquet wavenumber becomes real from Eq. (A8),

this case represents the outside of the band-gaps (passing

bands).

In the case of (b), Eq. (A2) is reduced to

a ¼ 0; if Re½g� ¼ 2

p; if Re½g� ¼ �2;

(A9)

b ¼ 0: (A10)

This one hence corresponds to the boundaries between the

passing band and the single band-gap.

In the case of (c), Eq. (A2) is reduced to

a ¼ 0; if Re½g� > 2

p; if Re½g� < �2;

(A11)

b ¼ lnjRe g½ �j

26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRe g½ �

2

� �2

� 1

s24

35 6¼ 0: (A12)

2808 J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa

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Since b 6¼ 0, this case corresponds to the single band-gap and

Eq. (17) for a¼ b can be obtained from Eqs. (A1) and

(A11).

In the case of (d), the following relations are obtained

from Eq. (A2)

a 6¼ 0 and a 6¼ p; (A13)

b 6¼ 0: (A14)

From the property shown by Braga and Herrmann34 that if Xis a solution of Eq. (15), so are its reciprocal 1/X as well as

its complex conjugate X*, it can be shown that if g is a solu-

tion of Eq. (16), so is its complex conjugate g*. Note that the

above property was originally proved for the case where all

plies are perfectly bonded, but it still holds for the case

where the spring-type interlayer interfaces are incorporated.

If the x3 component of the Floquet wavenumber correspond-

ing to g is denoted by fa and given by the right-hand side of

Eq. (A1) as fa¼ (2 mpþ aþ ib)/hu, then one corresponding

to g* and for the Floquet wave propagating in the same x3

direction fb can be written from Eq. (A2) as

fb ¼ ð2lp� aþ ibÞ=hu; (A15)

where l is an integer. From Eqs. (A13) and (A14), this case

corresponds to the double band-gap, and Eq. (17) for a 6¼bcan be obtained.

1S. I. Rokhlin, D. E. Chimenti, and P. B. Nagy, Physical Ultrasonics ofComposites (Oxford University Press, New York, 2011), pp. 225–368.

2M. F. Markham, “Measurement of the elastic constants of fibre composites

by ultrasonics,” Composites 1, 145–149 (1970).3R. E. Smith, “Ultrasonic elastic constants of carbon fibers and their

composites,” J. Appl. Phys. 43(6), 2555–2561 (1972).4B. Castagnede, J. T. Jenkins, W. Sachse, and S. Baste, “Optimal determi-

nation of the elastic constants of composite materials from ultrasonic

wave-speed measurements,” J. Appl. Phys. 67(6), 2753–2761 (1990).5E. P. Papadakis, T. Patton, Y.-M. Tsai, D. O. Thompson, and R. B.

Thompson, “The elastic moduli of a thick composite as measured by ultra-

sonic bulk wave pulse velocity,” J. Acoust. Soc. Am. 89(6), 2753–2757

(1991).6C. Arist�egui and S. Bastle, “Optimal recovery of the elasticity tensor of

general anisotropic materials from ultrasonic velocity data,” J. Acoust.

Soc. Am. 101(2), 813–833 (1997).7S. I. Rokhlin and W. Wang, “Double through-transmission bulk wave

method for ultrasonic phase velocity measurement and determination of

elastic constants of composite materials,” J. Acoust. Soc. Am. 91(6),

3303–3312 (1992).8Y. C. Chu and S. I. Rokhlin, “Comparative analysis of through-

transmission ultrasonic bulk wave methods for phase velocity measure-

ments in anisotropic materials,” J. Acoust. Soc. Am. 95(6), 3204–3212

(1994).9Y. C. Chu and S. I. Rokhlin, “A method for determination of elastic con-

stants of a unidirectional lamina from ultrasonic bulk velocity measure-

ments on [0/90] cross-ply composites,” J. Acoust. Soc. Am. 96(1),

342–352 (1994).10B. Hosten, “Stiffness matrix invariants to validate the characterization of

composite materials with ultrasonic methods,” Ultrasonics 30(6), 365–371

(1992).11B. Hosten, D. A. Hutchins, and D. W. Schindel, “Measurement of elastic

constants in composite materials using air-coupled ultrasonic bulk waves,”

J. Acoust. Soc. Am. 99(4), 2116–2123 (1996).12B. Hosten, M. Deschamps, and B. R. Tittmann, “Inhomogeneous wave

generation and propagation in lossy anisotropic solids. Application to the

characterization of viscoelastic composite materials,” J. Acoust. Soc. Am.

82(5), 1763–1770 (1987).

13B. Hosten, “Reflection and transmission of acoustic plane waves on an

immersed orthotropic and viscoelastic solid layer,” J. Acoust. Soc. Am.

89(6), 2745–2752 (1991).14S. Baudouin and B. Hosten, “Immersion ultrasonic method to measure

elastic constants and anisotropic attenuation in polymer-matrix and

fiber-reinforced composite materials,” Ultrasonics 34, 379–382

(1996).15S. Baudouin and B. Hosten, “Comparison between prediction and mea-

surement of viscoelastic moduli in composite materials versus temperature

using ultrasonic immersion technique with oil,” J. Acoust. Soc. Am.

102(6), 3450–3457 (1997).16M. Castaings, B. Hosten, and T. Kundu, “Inversion of ultrasonic, plane-

wave transmission data in composite plates to infer viscoelastic material

properties,” NDT&E Int. 33, 377–392 (2000).17D. E. Chimenti and D. Fei, “Scattering coefficient reconstruction in plates

using focused acoustic beams,” Int. J. Solids Struct. 39, 5495–5513

(2002).18N. Sela, O. Ishai, and L. Banks-Sills, “The effect of adhesive thickness on

interlaminar fracture toughness of interleaved CFRP specimens,”

Composites 20(3), 257–264 (1989).19D. H. Grande, L. B. Ilcewicz, W. B. Avery, and W. D. Bascom, “Effects

of intra- and inter-laminar resin content on the mechanical properties of

toughened composite materials,” in Proceedings of First NASA AdvancedComposite Technology Conference (1991), Vol. 2, 455–475.

20L. Wang and S. I. Rokhlin, “Ultrasonic wave interaction with multidirec-

tional composites: Modeling and experiment,” J. Acoust. Soc. Am.

114(5), 2582–2895 (2003).21A. Pilarski and J. L. Rose, “A transverse-wave ultrasonic oblique-

incidence technique for interfacial weakness detection in adhesive bonds,”

J. Appl. Phys. 63(2), 300–307 (1988).22S. I. Rokhlin and Y. J. Wang, “Analysis of boundary conditions for elastic

wave interaction with an interface between two solids,” J. Acoust. Soc.

Am. 89(2), 503–515 (1991).23A. Bostr€om and G. Wickham, “On the boundary conditions for ultrasonic

transmission by partially closed cracks,” J. Nondestruct. Eval. 10(4),

139–149 (1991).24S. I. Rokhlin and W. Huang, “Ultrasonic wave interaction with a thin ani-

sotropic layer between two anisotropic solids: Exact and asymptotic-

boundary-condition methods,” J. Acoust. Soc. Am. 92(3), 1729–1742

(1992).25P. Fraisse, F. Schmit, and A. Zarembowitch, “Ultrasonic inspection of

very thin adhesive layers,” J. Appl. Phys. 72(8), 3264–3271 (1992).26M. V. Golub, “Propagation of elastic waves in layered composites with

microdefect concentration zones and their simulation with spring bound-

ary conditions,” Acoust. Phys. 56(6), 848–855 (2010).27M. V. Golub and A. Bostr€om, “Spring boundary conditions and modeling

of 2D wave propagation in composites with imperfect interfaces,” in

Proceedings of International Conference Days on Diffraction (2010), pp.

73–78.28Y. Lu and J. D. Achenbach, “Effects of random deviations in interface

properties on the propagation of ultrasound in thick composites,”

J. Acoust. Soc. Am. 90(5), 2576–2585 (1991).29Y. Lu, “Effects of random fluctuations in ply mechanical properties on

ultrasound propagation in a laminated solid layer,” Ultrasonics 30(5),

289–295 (1992).30Y. Ishii and S. Biwa, “Ultrasonic evaluation of interlayer interfacial stiff-

ness of multilayered structures,” J. Appl. Phys. 111, 084907 (2012).31Y. Ishii and S. Biwa, “Evaluation of interlayer interfacial stiffness and

layer wave velocity of multilayered structures by ultrasonic spectroscopy,”

J. Acoust. Soc. Am. 136(1), 183–191 (2014).32L. Wang and S. I. Rokhlin, “Stable reformulation of transfer matrix

method for wave propagation in layered anisotropic media,” Ultrasonics

39, 413–424 (2001).33S. I. Rokhlin and L. Wang, “Stable recursive algorithm for elastic wave

propagation in layered anisotropic media: Stiffness matrix method,”

J. Acoust. Soc. Am. 112(3), 822–834 (2002).34A. M. B. Braga and G. Herrmann, “Floquet waves in anisotropic periodi-

cally layered composites,” J. Acoust. Soc. Am. 91(3), 1211–1227

(1992).35C. Potel, J.-F. Belleval, and Y. Gargouri, “Floquet waves and classical

plane waves in an anisotropic periodically multilayered medium:

Application to the validity domain of homogenization,” J. Acoust. Soc.

Am. 97(5), 2815–2825 (1995).

J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa 2809

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.54.110.31 On: Fri, 04 Mar 2016 04:52:13

36L. Wang and S. I. Rokhlin, “Floquet wave homogenization of periodic ani-

sotropic media,” J. Acoust. Soc. Am. 112(1), 38–45 (2002).37W. T. Thomson, “Transmission of elastic waves through a stratified solid

medium,” J. Appl. Phys. 21, 89–93 (1950).38N. A. Haskell, “The dispersion of surface waves on multilayered media,”

Bull. Seismol. Soc. Am. 43, 17–34 (1953).

39A. H. Nayfeh, “The general problem of elastic wave propagation in multi-

layered anisotropic media,” J. Acoust. Soc. Am. 89(4), 1521–1531 (1991).40B. Hosten and M. Castaings, “Transfer matrix of multilayered absorbing

and anisotropic media. Measurements and simulations of ultrasonic wave

propagation through composite materials,” J. Acoust. Soc. Am. 94(3),

1488–1495 (1993).

2810 J. Acoust. Soc. Am. 138 (5), November 2015 Yosuke Ishii and Shiro Biwa

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.54.110.31 On: Fri, 04 Mar 2016 04:52:13